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Hodge theory for combinatorial geometries
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron,
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the
Log-concavity of characteristic polynomials and the Bergman fan of matroids
In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing
Lorentzian polynomials
We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a
Rota's Conjecture and Positivity of Algebraic Cycles in Permutohedral Varieties.
you have taught will be invaluable for the remainder of my journey. Special thanks are due to my friend and collaborator Eric Katz. A large part of this thesis is based on our joint work. Lastly, I
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $|E|$ lines,
COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS
  • June Huh
  • Mathematics
    Proceedings of the International Congress of…
  • 30 November 2017
Why do natural and interesting sequences often turn out to be log-concave? We give one of many possible explanations, from the viewpoint of "standard conjectures". We illustrate with several examples
The maximum likelihood degree of a very affine variety
  • June Huh
  • Mathematics
    Compositio Mathematica
  • 3 July 2012
Abstract We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s
Varieties with maximum likelihood degree one
We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood
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